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Winning a Super Bowl Grid Pool: Frequency of Score Combinations in the NFL

It has come to my attention recently that amidst the fun of attending Super Bowl parties and watching the 2nd-most viewed sporting event on earth there are some people—seedy characters with questionable pasts, I'm sure—who are betting on the game! 

Now, as gambling on sporting events is highly regulated and illegal in almost every state, I'm confident that reports of this are overblown and that the fine, upstanding readers of this blog are not among those taking part in such an activity.

But if you happen to live in an area where such things are legal and you choose to participate, then you might be familiar with the kind of wager I want to write about today—the "Grid Pool."  Now the grid pool looks something like this before any entries are taken:

Blank Grid

Each participant pays a set amount for each square they would like, and once the grid is filled the two teams in the Super Bowl are randomly drawn to represent each axis, and the numbers 0-9 are randomly drawn to fill each row and column.  A completed bracket (in this case, of course, completely fictional) looks something like this:

Completed Grid

Because the teams and numbers are not entered until after the bracket is filled, participants have no control over which combinations of teams and numbers they get.

There are variations that use the score at the end of each quarter, but for simplicity we're going to use the jackpot—the score at the end of the game. The last digit on each team's score is used to find the pool winner...so in the above (fictional, I repeat, fictional) example, "Viv" would have won the pool as the final score was Ravens 34, 49ers 31, and "Viv" has the block corresponding to Ravens 4, 49ers 1:

Winning Grid

 

So, Nevada residents who have entered into one of these pools in a registered casino and intend on paying taxes on your winnings and complying with all other local, state, and federal laws, what combination of numbers should you be hoping for?

First off, since the teams are selected randomly there is no odds difference (initially) between squares that correspond to the combination of 1-4 and 4-1.  So I'm just going to "fold" the grid along the diagonal for simplicity.  I've pulled the results of every NFL game over the past five years—a total of 1,334 games—and compiled the number of times each combination has occurred. A cross-tabulation done in Minitab Statistical Software yields this output:

Tabulated Results

As an example, the combination 1-0 (which also includes 0-1) has occurred 28 times over the past five years.  Now tabulated results aren't the easiest to interpret so I'll show things graphically below, but you can tell (and statistics prove) that combinations occur with very different frequency—for example, 2-2 has only occurred once, while 3-0 has occurred 97 times.  If you happen to draw the 2-2 square, I hope you at least enjoy watching the game and commercials.

Let's use a Pareto Chart to look at the most common combinations, grouping the least common 20% into "Other":

Pareto Chart of Combination

As you can see, there are three clear leaders: drawing the 3-0, 7-4, or 7-0 squares (or their 0-3, 4-7, or 0-7 counterparts) is highly desirable. Although these six squares account for only 6% of the total number of squares, they account for nearly 22% of final scores!

Now the analysis above assumes that you don't know the winner with much certainty (sure, there's a favorite, but only really heavy favorites are almost certain to win), so which team gets drawn on which axis is irrelevant.  But there are two situations where you may not want to employ "folding" the grid for odds:

  1. One team is an overwhelming favorite (like the Broncos playing the Jaguars).
  2. The game is already underway and one team is obviously going to easily defeat the other, but it's too early in the game to have any reasonable idea of where final scores might end up.

In these cases, rather than folding we're going to look at the combination of the winner's score and loser's score.  Here our graph looks something like this, with winner's score listed first:

Pareto Chart of WL Combination

Now we see one combination that is clearly superior: having the square corresponding to 3 for the obvious winner and 0 for the obvious loser is the best.  To be sure, a combination like Winner 0 - Loser 7 is still much better than Winner 5 - Loser 4.  But if I could pick—hypothetically speaking, of course—I'd take 3-0, just 1% of the grid accounting for 5% of the results!

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